Problem: Find the sum of the geometric series $1 + 0.8 + 0.8^2 +0.8^3 +... + 0.8^{19}$ Choose 1 answer: Choose 1 answer: (Choice A) A $ 0.2 $ (Choice B) B $0.56$ (Choice C) C $ 4.94 $ (Choice D) D $4.98$
Getting started We're dealing with a geometric series because each term is multiplied by $0.8$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {1})$ is given in the question. The number of terms $n$ is ${20}$ because there are ${20}$ numbers from $0$ to $19$. [Where do the 0 and 19 come from?] The common ratio $r$ is ${0.8}$ because each term is multiplied by ${0.8}$ to get the next term. [How did we find the common ratio r?] Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{20}}&=\dfrac{{1}(1-\left({0.8}\right)^{{20}})}{1-\left({0.8}\right)} \\\\ S_{{20}}&=5(1-\left({0.8}\right)^{{20}})\\\\ S_{{{20}}} &\approx 4.94 \end{aligned}$ The answer $ 4.94 $